simple Lie algebra
Just as prime numbers are the multiplicative atoms of arithmetic, simple Lie algebras are the indivisible atoms out of which all semisimple Lie algebras are built. A simple Lie algebra cannot be broken into smaller ideals; it has no internal structure to quotient away. The astonishing fact is that, over the complex numbers, there are only a handful of these atoms, and they are completely classified.
A Lie algebra L is simple if it is nonabelian and has no ideals other than 0 and L itself. The nonabelian clause excludes the 1-dimensional algebra (which has no proper ideals but is too trivial to count). Simplicity forces [L, L] = L, so a simple Lie algebra has no abelian quotients at all. A semisimple Lie algebra is precisely a finite direct sum of simple ones.
Over an algebraically closed field of characteristic zero (such as C), the simple Lie algebras are classified by connected Dynkin diagrams: the classical families A_n = sl(n+1), B_n = so(2n+1), C_n = sp(2n), D_n = so(2n), and the five exceptionals G_2, F_4, E_6, E_7, E_8. This is one of the deepest and most beautiful classification theorems in mathematics, due to Killing and Cartan.
sl(2, C), spanned by e, f, h with [e, f] = h, [h, e] = 2e, [h, f] = -2f, is the smallest simple Lie algebra (type A_1, dimension 3). Any nonzero ideal must contain h, then e and f, hence all of sl(2): no proper nonzero ideal exists.
sl(2, C): the smallest simple Lie algebra, type A_1.
Simple as a Lie algebra (no proper ideals) is the infinitesimal cousin of a simple group (no proper normal subgroups). The two notions are linked: the Lie algebra of a simple Lie group is simple, up to the center subtlety.